The Classification of Real Purely Infinite Simple C*-Algebras

We classify real Kirchberg algebras using united K-theory. Precisely, let A and B be real simple separable nuclear purely infinite C*-algebras that satisfy the universal coefficient theorem such that A C and B C are also simple. In the stable case, A and B are iso-morphic if and only if K CRT (A) ∼ = K CRT (B). In the unital case, A and B are isomorphic if and only if (K CRT (A), [1 A ]) ∼ = (K CRT (B), [1 B ]). We also prove that the complexification of such a real C*-algebra is purely infinite, resolving a question left open from [43]. Thus the real C*-algebras classified here are exactly those real C*-algebras whose complexification falls under the classification result of Kirchberg [26] and Phillips [35]. As an application, we find all real forms of the complex Cuntz algebras O n for 2 ≤ n ≤ ∞.


Introduction
One of the highlights of the classification theory of simple amenable C*-algebras is the classification of purely infinite nuclear simple C*-algebras, obtained by Kirchberg and Phillips in [26] and [35].This classification theorem relies in an essential way on the Universal Coefficient Theorem established by Rosenberg and Schochet in [40], where it was observed that "For reasons pointed out already by Atiyah, there can be no good Künneth Theorem or Universal Coefficient Theorem for the KKO groups of real C*-algebras; this explains why we deal only with complex C*-algebras".Thus at the time of the Kirchberg and Phillips classification, the lack of a universal coefficient theorem was the primary barrier to extending the classification result to real C*-algebras.However, in [8], a new invariant called united K-theory was introduced for real C*-algebras and in [9] a universal coefficient theorem was proven for real C*-algebras using united K-theory.In the present paper, we take advantage of these developments to provide a complete classification of a class of real simple purely infinite C*-algebras in terms of united K-theory.The real C*-algebras that are classified are exactly those real C*-algebras for which the complexification is covered by the Kirchberg and Phillips theory.As an application of our classification we determine all the real forms of the complex Cuntz algebras O n for 1 ≤ n ≤ ∞: there are two such forms when n is odd and one when n is even or infinite.The overall framework of the proof will be the same as that in the paper [35] and the underlying theory on which that paper was built.Furthermore, many of the proofs in the development leading to the main theorems of [35] carry over to the real case without significant change.In those cases, we will simply refer to the established proofs in the literature without reproducing them here.However there are many situations where the arguments in the real case require modification and we will then provide full proofs or full discussion of the necessary modifications.In Section 2, we describe the invariant of united K-theory and summarize its key properties.In Section 3 we then establish real analogues of some of the fundamental properties of purely infinite algebras, in the course of which we resolve a problem left hanging in [43] and [13] by showing that the complexification of a purely infinite simple real C*-algebra is also purely infinite (using the original definition for simple algebras).Following the complex case, as developed in [38], we then establish (in Theorem 5.2) criteria for two unital homomorphisms from the real Cuntz algebra O R n (n even) to be approximately unitarily equivalent.Modifications of the complex arguments are required to establish some of the preliminary results: in Section 4 we modify the required results about exponential rank, noting that the close link between self-adjoint and skew-adjoint elements is absent in a real C*-algebra, and in Section 5 we modify the result from [15] establishing the Rokhlin property of the Bernoulli shift on the CAR-algebra.Our next step is to establish real analogues of Kirchberg's tensor product theorems and his embedding theorem.This is achieved in Section 6 by using the relevant complex results and the embedding of C into M 2 (R).In Sections 7, 8, 9 and 10, we closely follow [35] indicating how the results achieved for the complex case can be obtained in the real case.In particular, Section 7 contains a key result about uniqueness of homomorphisms from O R ∞ to a real purely infinite C*-algebras.Section 8 contains the theory of asymptotic morphisms in the context of real C*-algebras and Section 9 culminates in a theorem identifying KK-theory to a group of asymptotic unitary equivalence classes of asymptotic morphisms as in Section 4 of [35].To accomplish this, we make use of the axiomatic characterization of KK-theory for real C*-algebras established in [12].This development culminates in Section 10, which contains the statements and proofs of our classification theorems, and in Section 11, which uses these results to describe the real forms of Cuntz algebras.The notation we use in these sections closely follows that in [35].
We will use the notation H R for a real Hilbert space; and B(H R ) and K R for the real C*-algebras of bounded and compact operators H R .For the complex versions of these objects we will use H, B(H), and K.For a C*-algebra A, we will write M n (A) for the matrix algebra over A; and M n will stand for M n (R).Following standard convention, we will use O n for the complex Cuntz algebras and O R n for the real versions.The complexification of a real C*-algebra A will be denoted by A C .We will use Φ throughout to denote the conjugate linear automorphism of A C defined by a + ib → a − ib (for a, b ∈ A).Note that A can be recovered from Φ as the fixed point set.Finally, a tensor product written as A ⊗ B will in most cases be the C*-algebra tensor product over R, but should be understood to be a tensor product over C if both A and B are known to be complex C*-algebras.Recall that if A and B are real C*-algebras, then

Preliminaries on United K-Theory
United K-theory was developed in the commutative context in [14] and subsequently extended to the context of real C*-algebras in [8].United K-theory consists of the three separate K-theory modules as well as several natural transformations among them.In this section, we give the definition of united K-theory and summarize the features needed in this paper.Details are in [8], [9], [10].
In this definition, KO * (A) = K * (A) is the standard K-theory of a real C*algebra, considered as a graded module over the ring K * (R).This means in particular that there are operations corresponding to multiplication by the elements of the same name in KO * (R).The operation β O is the periodicity isomorphism of real K-theory.The second item KU * (A) = K * (A C ) is the K-theory of the complexification of A, having period 2. It is a module over K * (C), which is to say that that there is an isomorphism of period 2 and the two remaining groups are independent with no operations between them.Finally, KT * (A) is the period 4 self-conjugate K-theory originally defined in the topological setting in [1].In the non-commutative setting, it is more easily defined as KT * [8]).Self-conjugate K-theory is a module over the ring K * (T ), giving operations The rest of the information in united K-theory consists of operations among the three K-theory modules.For example, c is induced by the natural inclusion A → A C ; r by the inclusion A C → M 2 (A); and ψ U by the involution Φ on A C .These operations are known to satisfy the following relations (see Proposition 1.7 of [8]): United K-theory takes values in the algebraic category of CRT-modules.A CRT-module consists of a triple (M O , M U , M T ) of graded modules, one over each of the rings K * (R), K * (C), and K * (T ); as well as natural transformations c, r, ε, ζ, ψ U , ψ T , γ, τ that satisfy the above relations.
For any real C*-algebra A, the CRT-module K CRT (A) is acyclic, which means that the sequences The important advantage of the full united K-theory over ordinary K-theory for a real C*-algebra A is that it yields both a Künneth formula (Theorem 4.2 of Documenta Mathematica 16 (2011) 619-655 [8]) and a univeral coefficient theorem (Theorem 1.1 of [9]).For later reference, we now state two results that follow from those fundamental theorems.

The unital inclusion R → O R
∞ induces an isomorphism on united K-theory.This follows from Theorem 4 of [10] and the fact that the unital inclusion C → O ∞ induces an isomorphism on (complex) K-theory.Thus, Theorem 3.5 of [8] gives Then the isomorphism of (2) follows by the Main Theorem of [8].
Recall from [41] that the bootstrap class N is the smallest subcategory of complex, separable, nuclear C*-algebras that contains the separable type I C*algebras; that is closed under the operations of taking inductive limits, stable isomorphisms, and crossed products by Z and R; and that satisfies the two out of three rule for short exact sequences (i.e. if 0 → A → B → C → 0 is exact and two of A, B, C are in N , then the third is also in N ).
This last result is the essential preliminary result for our classification of real purely infinite simple C*-algebras.We will also make use of Theorem 1 of [10], which states that every countable acyclic CRT-module can be realized as the united K-theory a real separable C*-algebra, indeed the C*-algebra can even be taken to be simple and purely infinite.We now describe a simpler variation of united K-theory that, by results from [23], contains as much information as the full version of united K-theory.Definition 2.4.Let A be a real C*-algebra.Then For any real C*-algebra, K CR (A) is an acyclic CR-module, which means that the relations are satisfied and that the sequence Let Γ be the forgetful functor from the category CRT-modules to the category of CR-modules.It is immediate from Theorem 4.2.1 of [23] that Γ is injective (but not surjective) on the class of acyclic CRT-modules.Hence we have the following result.
Proposition 2.5.Let A and B be real C*-algebras.Then Note, however, that the results of [10] do not extend to CR-modules.Not every countable acyclic CR-module can be realized as K CR (A) for a real C*-algebra A.

Preliminaries on Real Simple Purely Infinite C*-Algebras
In this section, we provide some preliminaries on simple and purely infinite C*-algebras, including a theorem characterizing simple purely infinite real C*algebras in terms of their complexification.One direction of this characterization was achieved in [43] and [13].
Let A be a real unital C*-algebra, let U(A) denote the group of unitary elements in A, and let U 0 (A) denote the connected component of the identity in U(A).
Note that if u is a unitary in a real C*-algebra, then the spectrum σ(u) ⊆ T satisfies σ(u) = σ(u) and the real C*-algebra generated by u is isomorphic to the algebra of complex-valued continuous functions f on σ(u) that satisfy As the structure of simple complex C*-algebras is comparatively wellunderstood, our primary interest lies in c-simple C*-algebras.
As in the complex case, we will use the tilde ∼ to denote the relation of Murrayvon Neumann equivalence of projections.A projection is said to be infinite if it is Murray-von Neumann equivalent to a proper subprojection of itself.The following definition of purely infinite is from [43].Bearing in mind subsequent developments, such as [27] and [28], a different definition should be made in the non-simple case.However the focus in this paper is on simple algebras, for which the definition below is appropriate.(1) A subalgebra B is a regular hereditary subalgebra of A if there is an element x ∈ A + such that B = xAx.(2) A is purely infinite if every regular hereditary subalgebra of A contains an infinite projection.
Proposition 3.4.Let A be a separable simple purely infinite real C*-algebra.Then either A is unital or there is a real unital simple purely infinite C*-algebra Proof.As in Section 27.5 of [2].
Proposition 3.5.Let A be a simple purely infinite C*-algebra and let p be a projection in A. Then pAp and A are stably isomorphic.
Proof.In the complex case, this result follows from Corollary 2.6 of [16].The proof of that result and the proofs of the preliminary lemmas of Section 2 of [16] work the same in the real case.
For the rest of this section, f ε will denote the real-valued function such that Lemma 3.6.For any real C*-algebra A, the following are equivalent.
(1) For any non-zero a, b ∈ A there exist x, y ∈ A with a = xby.
(2) For any non-zero positive a, b ∈ A there exists x ∈ A with a = xbx * . Proof.
(1) ⇒ (2).This uses the argument for the complex case, from Lemma 1.7 and Proposition 1.10 of [18].If a, b ∈ A are positive and non-zero and ε is chosen so that f ε (b) = 0 then a = (zz * zk)b(zz * zk) * , where x, y are chosen so that Lemma 3.7.Let A be a real C*-algebra such that for all non-zero elements a, b there exist x, y with a = xby.Suppose that A contains a non-zero projection and let c be a non-zero positive element such that cAc = A. Then cAc contains an infinite projection.
Proof.The argument from (vii) ⇒ (i) of Theorem 2.2 of [31] applies to the real case to show that for any non-trivial projection p and positive element x there is a Murray-von Neumann equivalence between p and a subprojection of x.We will repeatedly use this fact.
In the unital case, this shows that the unit 1 is Murray-van Neumann equivalent to a projection of cAc, which is necessarily infinite.Now suppose that A has no unit but has a non-zero projection p. Applying the fact above to a non-zero positive element d in (1 − p)A(1 − p) gives a projection q such that p ∼ q and p ⊥ q.Now apply the fact again using the projection p + q and the positive element p to show that p + q is infinite.Finally, apply the same fact using the projection p + q and the positive element c to show that p + q is Murray-von Neumann equivalent to a projection in cAc.
Lemma 3.8.Let A be a real simple C*-algebra.Then the following are equivalent: (1) A is purely infinite, (2) A is not isomorphic to R, C, or H and for each pair of non-zero elements a, b ∈ A there exist x, y ∈ A such that a = xby, (3) A is not isomorphic to R, C, or H and for each pair of non-zero positive elements a, b ∈ A there exists x ∈ A such that a = xbx * .Furthermore, if these conditions are satisfied, then for all ε > 0 the element x in (3) can be chosen to satisfy x ≤ ( a / b ) 1/2 + ε.
Proof.As the result is well-known in the complex case, we may assume by Theorem 3. Proof.From Theorem 3.3 of [43] we know that A is purely infinite if A C is.For the converse, suppose A is purely infinite, let ω be a free ultrafilter on N and let A ω be the corresponding ultrapower algebra, defined in Definition 6.2.2 of [39].Note that the proofs of Proposition 6.2.6 of [39] and the preliminary Lemma 6.2.3 carry over directly to the real case (using Lemma 3.8).Therefore A ω is simple and purely infinite.Suppose that D is a dimension function, as defined in Definition I. is weakly purely infinite by Theorem 4.8 of [28].By Corollary 4.16 of [28] it is therefore purely infinite.Proof.These results follow immediately from Theorem 3.9 and the same results in the complex case (see Proposition 4.1.8 of [39]).
We now work toward showing that the K 0 and K 1 groups of a real purely infinite algebra can be described in a similar way to the complex case.The next two lemmas provide the required modification of Lemma 1.7 of [19].
Lemma 3.11.Let A be a real c-simple purely infinite unital C*-algebra and let u ∈ U(A) and let λ ∈ σ(u).For any ε > 0 there exists v ∈ U(A) such that u − v < ε and Proof.First assume that λ = λ * .Let h be a positive function on σ(u) such that supp(h) ⊂ N ε0 (λ) and h(z * ) = h(z) for all z ∈ σ(u).Then h(u) ∈ A and let p be a non-zero projection in h(u)Ah(u).As in the proof of Lemma 1.7 of [19], we have u − (p ⊥ up ⊥ + λp) ≤ 3ε 0 .Then the polar decomposition of (p ⊥ up ⊥ + λp) yields a unitary v of the required form that, if ε 0 is sufficiently small, will satisfy u − v < ε.Now assume λ = λ * .Choose ε 0 small enough so that N ε0 (λ) ∩ N ε0 (λ * ) = ∅.Let h 1 be a positive function on σ(u) such that supp(h 1 ) ⊂ N ε0 (λ).By Theorem 3.9, A C is purely infinite so there is a non-zero projection Thus p 1 and p 2 are orthogonal projections and p ∈ A.
As in Lemma 1.7 of [19], we have Lemma 3.12.Let A and u be as above.Then there is a projection p in A and a unitary v in U(p ⊥ Ap ⊥ ) such that u ∼ v + p.
Proof.If 1 ∈ σ(u) then using Lemma 3.11, approximate u by an element of the form v + p.If the approximation is close enough, then the two unitaries will be in the same path component.If λ ∈ σ(u) where λ = λ * , use Lemma 3.11 to approximate u by v + λp 1 + λ * p 2 .
Then we can easily find a path from The only possibility left is u = −1.In that case, find two orthogonal projections q 1 and q 2 and a partial isometry s such that ss * = q 1 and s * s = q 2 .Let p = q 1 + q 2 .The projection p can be rotated to −p within the 2 × 2 matrix algebra generated by q 1 , q 2 and s.Hence the unitary −1 = −(p ⊥ ) + −p can be connected to the unitary −(p ⊥ ) + p.
Proposition 3.13.Let A be a c-simple purely infinite real C*-algebra.Then Proof.In the complex case, these results are proven in Section 1 of [19].The proofs of those results as well as the proofs of the preliminary lemmas carry over to the real case, with two modifications.The first is to the proof of Lemma 1.7 of [19], which we already addressed with the proof of Lemma 3.12 above.Secondly, in the proof of Lemma 1.1 of [19] the author uses an element of the form that is a unitary lying in the finite dimensional C*-algebra generated by w.In the complex case it follows that w ∈ U 0 (A), whereas in the real case unitary groups of finite dimensional C*-algebras are not connected in general.However, if instead we take w = w − w * + (1 − w * w − ww * ) then w is in the connected component of the identity, as it corresponds to a matrix of the form 0 1 −1 0 .The proof of Lemma 1.1 of [19] can be completed without change using this alternative w.
We note that part (1) of Proposition 3.13 appeared as Proposition 11 of [10].
If a is skew-adjoint, then σ(a) = −σ(a) ⊆ iR and the real unital C*-algebra generated by a is isomorphic to Furthermore, if a is a skew-adjoint element in a real unital C*-algebra A, then exp(a) is a unitary in A.

Documenta Mathematica 16 (2011) 619-655
Proof.Suppose first that u is a unitary element in A with u − 1 < 2. Then −1 / ∈ σ(u).We define a continuous function f : T \ {0} → i(−π, π) by f (exp(it)) = it for t ∈ (−π, π).Then f (u) is in the real C*-algebra generated by u, is skew-adjoint, and satisfies exp(f (u)) = u.More, generally, if u ∈ U 0 (A) then there exists a chain Then applying the previous paragraph we have The argument in the proof above also implies that the set E n+1 contains the topological closure of E n so that we have the an increasing sequence similar to that in [37], motivating the following definition.Definition 4.3.
(1) The exponential rank of A, written cer(A), is equal to the integer n if E n is the smallest set in this sequence to be equal to U 0 (A) and is equal to the symbol n + ε if E n is the smallest set to be equal to U 0 (A).If E n = U 0 (A) for all n then cer(A) = ∞.(2) The exponential length of A, written cel(A), is equal to the smallest number 0 < cel(A) ≤ ∞ such that every unitary u in U 0 (A) can be written in the form where k i ∈ A sk and With these definitions, the proofs of Section 2 of [37] can be applied with minimal modification to prove the following results.
Lemma 4.4.Let A be a real unital C*-algebra and let n be a positive integer.
(2) If cel(A) ≤ nπ then cer(A) ≤ n + ε.In the case of a complex C*-algebra A there is a bicontinuous bijection A sa → A sk given by multiplication by i, showing that A has skew rank zero if and only it has real rank zero.However, in the case of real C*-algebras things are more subtle.For example the condition of being skew-rank zero is not equivalent (in the unital case) to the condition that the invertible elements of A sk are dense.Indeed, all finite dimensional real C*-algebras have real skew rank zero, but the invertibles of (M n ) sk are dense only if n is even.Proof.Let A be a real purely infinite C*-algebra such that [1] ∈ 2K 0 (A).Let a ∈ A sk and let ε > 0 be given.Define functions g : iR → R and f : iR → iR by For the second statement, again let a ∈ A sk and let ε > 0 be given.By the first part of the theorem, we may assume that a is invertible, hence σ(a) ⊂ iR \ {0}.Write a = a 1 + a 2 where the elements a i ∈ A C satisfy σ(a 1 ) ⊂ i(0, ∞) and σ(a 2 ) ⊂ i(−∞, 0).Note also that Φ(a 1 ) = a 2 .Since A C is simple and purely infinite it has real rank zero, so there exists Let u ∈ U(A) be a unitary such that σ(u) = S 1 .Then for every ε > 0 there is a unitary v with finite spectrum such that u − v < ε.
that is a right inverse to the function it → exp(it) and that satisfies f (z * ) = f (z) * .Then f (u) ∈ A sk can be approximated within δ by a skew adjoint element b with finite spectrum by Proposition 4.7.For an appropriate choice of δ, this implies that exp(b) ∈ U(A) approximates u within ε.Similarly, if 1 / ∈ σ(u), then there is a continuous function f : σ(u) → i[−π, π] that is a right inverse to the function it → − exp(−it) and that satisfies f (z * ) = f (z) * .In the general case, suppose that λ / ∈ σ(u) for some λ ∈ S 1 .Let , where E u (σ i ) denotes the spectral projection of u associated with the clopen subset σ i of σ.Then 1 / ∈ σ(u 2 ) and −1 / ∈ σ(u 1 ).Using the results from the first two paragraphs, let v i be a unitary that approximates a unitary that approximates u within ε.Lemma 4.9.Let A be a real unital simple purely infinite C*-algebra let u ∈ U(A) and let {λ 1 , . . ., λ n } be a subset of σ(u) that is closed under conjugation.For any ε > 0 there exist v ∈ U(A) and orthogonal projections p 1 , . . ., Furthermore, the elements n i=1 p i and n i=1 λ i p i are both in A. Proof.Use the constructions of Lemma 3.11 above as in the proof of Lemma 6 of [34].
Lemma 4.10.Let A be a real unital C*-algebra and let u ∈ U(A).For any ε > 0 there exists an h ∈ M 2 (A) sk such that u ⊕ u * − exp(h) < ε.
Proof.As in the proof of Corollary 5 of [34], there exists a continuous path v(t) of unitaries in M 2 (A) with v(0) = 1 and v(π/2) = u ⊕ u * such that −1 / ∈ σ(v(t)) for 0 ≤ t < π/2.Thus we can find a t close enough to π/2 such that u ⊕ u * − v(t) < ε and v(t) = exp(h) for a skew-adjoint h.Lemma 4.11.Let A be a real unital c-simple purely infinite C*-algebra such that [1] ∈ 2K 0 (A).Let e 1 , e 2 , e 3 , e 4 be nonzero orthogonal projections in A that sum to 1. Let a be a partial isometry such that a * a = e 2 and aa * = e 3 .Let u ∈ U(e 1 Ae 1 ) and v ∈ U(e 2 Ae 2 ) be unitaries with σ(u) = S 1 .Then for all ε > 0 there is a unitary z ∈ U(A) and a unitary w ∈ U(e 4 Ae 4 ) with finite spectrum such that Proof.This proof closely follows that of Lemma 7 of [34].By Lemma 4.10 there is a unitary in (e 2 + e 3 )A(e 2 + e 3 ) that is arbitrarily close to v + av * a * and that has the form exp h for h ∈ A sk .This in turn can be approximated by a unitary that has finite spectrum by Proposition 4.7.The general form of such a unitary is n k=1 where λ * k = λ k , the nonzero projections q ki ∈ A C satisfy Φ(q k1 ) = q k2 for 1 ≤ k ≤ n, and the (possibly zero) projections q 0i are in A. Furthermore, the q ki are orthogonal and sum to e 2 + e 3 .Without loss of generality, we assume that v + av * a * has this form.With an obvious choice of coefficients λ ki we can write this as (Henceforth in this proof will use an undecorated to represent a double sum indexed as n k=0 2 i=1 .)Now we replace u by a nearby element of the form given by Lemma 4.9.Specifically, there are orthogonal projections p ki ∈ e 1 A C e 1 and, setting p = e 1 − p ki ∈ A, there is a unitary u 0 ∈ pAp such that (where the projection p 0i = 0 if and only if q 0i = 0).
Theorem 4.12.Let A be a real unital c-simple purely infinite C*-algebra such that [1] ∈ 2K 0 (A).For every u ∈ U 0 (A) and every ε > 0 there is a unitary v with finite spectum such that u − v < ε.
Proof.With the lemmas that we have developed, the proof is now the same as that of the unital case of Theorem 1 and Corollary 2 of [34], except that wherever there is an element of the form exp(ih) where h is self-adjoint, we use exp(k) where k is skew-adjoint.
As in the complex case, we have the following corollary concerning exponential length.
Proof.By Theorem 4.12, every unitary u ∈ U 0 (A) can be approximated within ε by a unitary v with finite spectrum.For ε sufficiently small, v * u − 1 < ε implies there exists a skew-adjoint k 2 such that v * u = exp(k 2 ) with k 2 ≤ 4 − π.As v has finite spectrum, there exists a skew-adjoint k 1 such that v = exp(k 1 ) and k 1 ≤ π.Then u = exp(k 1 ) exp(k 2 ) and k 1 + k 2 ≤ 4.

Homomorphisms from O R n
The following theorem gives the real version of the Rokhlin property of the Bernoulli shift, established in [15] and summarized in [39].Let M 2 ∞ = lim k→∞ M 2 k be the real CAR algebra and let H be the real C*-algebra of quaternions.
where there are sufficiently many initial zeros to make f j orthogonal to its predecessors and where n j is chosen so that and Sf j − ω j f j < δ.If f j = g j + ih j with g j , h j ∈ ℓ 2 (N, R) then, from the orthogonality of f j and f j , g j = h j = 1/ √ 2. Let a : ℓ 2 (N, C) → A C be the map described in [15] and [39] satisfying the canonical anticommutation relations and observe that a maps ℓ 2 (N, R) It follows from this that the elements w i for 1 ≤ i ≤ 2r − 1 satisfy the same relations.Therefore, using the matrix units described in the proof of Proposition 4.1 of [15], the real C*-algebra B generated by w 1 , . . ., w 2r−1 is isomorphic to M 2 2r−1 .Slightly varying the proof of Proposition 4.1 of [15], let β be the automorphism of the complexification of B determined by , so that β leaves the real algebra B invariant.Identifying B with M 2 2r−1 , there is an orthogonal matrix W implementing β.By standard linear algebra, described for example in Section 81 of [22], W is orthogonally conjugate to an orthogonal matrix consisting of diagonal elements ±1 and diagonal 2 × 2 rotation matrices, determined by the eigenvalues of W .As in [39], on the complexification of B, identified with M 2 2r−1 (C), β is implemented by a diagonal unitary with entries 1, ω r , ω 2 r , . . ., ω 2 r −1 r , each repeated 2 r−1 times.(The unitary arises as the tensor product of one diagonal unitary with entries 1, ω r , ω 2 r , . . ., ω 2 r −1 r and another with entries 1, ω r , ω r 2 , . . ., ω r 2 r−1 −1 .)On B ∼ = M 2 2r−1 the orthogonal matrix W implementing β is therefore conjugate to an orthogonal matrix with 2 × 2 diagonal blocks diag(1, −1), R, R 2 , . . ., R 2 r−1 −1 , each repeated 2 r−1 times, where .
The cyclic shift on M 2 r is implemented by the unitary , which is orthogonally conjugate to diag diag(1, −1), R, R 2 , . . ., R 2 r−1 −1 .It follows that the orthogonal element W implementing β on B is orthogonally conjugate to a direct sum of 2 r−1 copies of V and thus that β is conjugate to a direct sum of 2 r−1 cyclic shifts.It follows that there are 2 r orthogonal projections e 0 , e 1 , . . ., e 2 r = e 0 in B (each of rank 2 r−1 ) that are cyclically permuted by β.As in the proof of Proposition 4.1 of [15], a suitable choice of δ at the start of the proof ensures that σ(e j ) − β(e j ) < ε for each j and therefore the projections e 0 , e 1 , . . ., e 2 r = e 0 have the required properties.
Let n be an even integer, let φ, ψ be unital homomorphisms from O R n to D, let λ be the endomorphism of D defined by λ(a) = n j=1 φ(s j )aφ(s j ) * and let u ∈ U(D) be defined by u = n j=1 ψ(s j )φ(s j ) * , where s 1 , . . ., s n are the canonical generators of O R n .Then the following are equivalent: φ and ψ are approximately unitarily equivalent.In particular, these statements are equivalent if D is a real unital purely infinite c-simple C*-algebra.
Proof of Theorem 5.2.The proof of the equivalence of the four statements, assuming (i) and (ii), is similar to that of the complex case in Sections 3 and 4 of [38], modified only by the use of unitaries of the form exp(h) with h ∈ A sk in the proof of the real version of Lemma 4.6 of [38].We note that in the proof of the real version of Lemma 3.7 of [38], the required result from [19] holds, as was observed already in the proof of Proposition 3.13 above.Suppose D is a real unital purely infinite c-simple C*-algebra.Then condition (i) holds for D by Proposition 3.13.Since K Using the unital homomorphism φ (or ψ) we obtain [1 D ] ∈ 2K 0 (D).Then condition (ii) holds by Corollary 4.13.

Tensor Product Theorems
In this section, we reproduce for real C*-algebras some standard results regarding tensor products with O R 2 and O R ∞ .Definition 6.1.
(1) A real (resp.complex) C*-algebra A is amenable if for all ε > 0 and all finite subsets F ⊂ A, there is a finite dimensional real (resp.complex) C*-algebra B and contractive completely positive linear maps φ : A → B and ψ : B → A such that ψ • φ(a) − a < ε for all a ∈ F .
(2) A real (resp.complex) C*-algebra A is nuclear if for all real (resp.complex) C*-algebras B the algebraic tensor product A ⊗ R B (resp. A ⊗ C B) has a unique C*-norm.(3) A real (resp.complex) C*-algebra A is exact if the tensor product functor B → A ⊗ min B is exact.Here the tensor product is over R (resp.C) and B can be any real (resp.complex) C*-algebra.Lemma 6.2.Let A be a real C*-algebra.Then (1) A is amenable if and only if A C is amenable.
(2) A is nuclear if and only if A C is nuclear.
(3) A is exact if and only if A C is exact.Consequently, A is amenable if and only if it is nuclear; and in this case it is also exact.
Proof.Part (1) can be found in Proposition 3 of [25] and the preceding text.We claim that there is a one-to-one correspondence between C*-norms on the algebraic tensor product A ⊗ R B and those on It also follows that the restriction of the minimal C*-norm on A C ⊗ C B C gives the minimal C*-norm on A ⊗ R B. This fact, plus the fact that the complexification functor A → A C is exact, implies (3).The final statement then follows from the corresponding statement for complex C*-algebras.See Theorem 6.1.3 of [39] and Theorem 6.5.2 of [32].Proposition 6.3.Let A be a real separable C*-algebra A. Then A is exact if and only if there is an injective homomorphism ι : If A is unital then ι can be chosen to be unital.
Proof.Suppose that A is exact.Then A C is separable and exact.Thus, by Theorem 6.3.11 of [39], there is an injective homomorphism ι C : A C → O 2 (which is unital if A C is unital).Then we can take ι to be the composition Conversely, if there is an injective homomorphism ι : A → O R 2 then the complexification yields an injective homomorphism from A C to O 2 .By Theorem 6.3.11 of [39] this implies that A C is exact, hence A is exact.Lemma 6.4.Let A be a real purely infinite c-simple nuclear unital C*-algebra.Then all unital endomorphisms on A ⊗ O R 2 are approximately unitarily equivalent.
Proof.In the complex case, this result is found as Theorem 6.3.8 of [39].We will use that result to prove the real version.By Corollary 5.3, Part (5) it suffices to show that any unital homomorphism is approximately unitarily equivalent to the identity.We write be the canonical injections.Then we use the commutative diagram By Theorem 6.3.8 of [39], there is a sequence of unitaries given such that a = 1 and let ε > 0. Then find an integer N large enough so that, for all n ≥ N , Then a calculation shows that, for all n ≥ N , Theorem 6.5.Let A be a real C*-algebra.Then A is c-simple, separable, unital, and nuclear if and only if 2 holds for a real C*-algebra A, then we have A C ⊗ O 2 ∼ = O 2 which implies by Theorem 7.1.2 of [39] that A C is simple, separable, unital, and nuclear.Therefore A is c-simple, separable, unital, and nuclear.
We note that the hypothesis above requiring that A be c-simple cannot be relaxed, as the result does not hold for A = O 2 (considered as a real C*algebra).Theorem 6.6.
(1) Any two unital homomorphisms φ, ψ from O R ∞ into a real, unital, purely infinite, nuclear, c-simple C*-algebra A are approximately unitarily equivalent.
(2) Let A be a real c-simple, separable, and nuclear C*-algebra.Then A is isomorphic to As in Section 7.2 of [39].Documenta Mathematica 16 (2011) 619-655 Corollary 6.7.Let A and B be real, c-simple, separable, nuclear C*-algebras.If A or B is purely infinite, then A ⊗ B is purely infinite.

∞
The goal of this section is to prove the following theorem, analogous to Proposition 2.2.7 of [35].The proof of Theorem 7.1 will be the same as that in [35].However, there are a couple of background topics that need to be addressed in the context of real C*-algebras.We begin with a discussion of approximately divisible real C*-algebras, following [6].It is sufficient to consider only separable unital C*-algebras.Also, we skirt the general topic of completely noncommutative C*-algebras by taking into account Definition 2.6 of [6] and the subsequent comment.Definition 7.2.A separable unital real C*-algebra A is approximately divisible if for all x 1 , x 2 , . . ., x n ∈ A and ε > 0, there is a unital subalgebra B isomorphic to M 2 , M 3 , or M 2 ⊕ M 3 such that x i y − yx i < ε for all i = 1, 2, . . ., n and all y in the unit ball of B.
The following theorem is the real version of Corollary 2.1.6 of [35].
Lemma 7.3.The tensor product O R ∞ ⊗ D is approximately divisible for any real separable unital C*-algebra D. In particular, every c-simple, separable, nuclear, purely infinite, unital real C*-algebra is approximately divisible.
Theorem 6.6 we obtain a sequence of mutually commuting unital homomophisms ∞ and let ψ n = φ n • γ.Then for large enough n, the subalgebra B = ψ n (M 2 ⊕ M 3 ) works.The second statement follows from part (2) of Theorem 6.6.
Lemma 7.4.Let p and q be full projections in M ∞ (A) where A is a real, separable, unital, approximately divisible C*-algebra.Then p ∼ q if and only if Proof.The proof is the same as the proof of (the first part of) Proposition 3.10 in [6] in complex case.That proof relies on a progression of results from Section 2 of [6] which can all be proven in the real case in the same way with one minor caveat.The proof of Proposition 2.1 of [6] (which in that paper was left to the reader) relies on the fact that a complex C*-algebra is spanned by its unitaries.While this fact is not true in general for real C*-algebras, it can easily be shown to be true for finite dimensional real C*-algebras, which is the relevant case.The proof of Proposition 3.10 in [6] also relies on Theorem 3.1.4of [3], which is a ring-theoretic result stated in enough generality to apply to real C*-algebras.
We remark that a more direct proof of Lemma 7.4 can be achieved in the special case (which is sufficient for our purposes) that A = O R ∞ ⊗ D where D is separable and unital.In that case, we write ∞ ⊗ D be the unital subalgebra of A consisting of the first n factors in the tensor product.Then for each n and each k, it is easy to find a unital subalgebra Thus we achieve the result of Corollary 2.10 of [6] without having to recheck all the earlier material of Section 2 of [6] in the real case.
Lemma 7.5.Let D be a unital real C*-algebra and let p, q be any two full Then p is Murray-von Neumann equivalent to a subprojection of q.Furthermore, p is homotopic to q if and only if they represent the same class in With our Lemmas 7.3 and 7.4, as well as Theorem 3.6 of [11], the proof is the same as that of Lemma 2.1.8 of [35].
Proof of Theorem 7.1.With these preliminary definitions and results, the proof is the same as the proof of Proposition 2.2.7 of [35] including all of the lemmas and intermediate results in Sections 2.1 and 2.2 of [35].We note that in [35], the proofs of Propositions 2.1.9and 2.1.10(having to do with exact stability of the relations defining O R m and E m (δ)) are referred back to the proofs of parts (1) and (2) of Lemma 1.3 of [30].The proof given there for part (2) produces isometries w j that live in the real algebra E n (δ).Therefore the homomorphisms φ (m) δ constructed in the complex case restrict to homomorphisms between the real algebras.The same will be true for the analogous proof of part (1).We also note that the proofs for the real versions of Lemmas 2.2.1 and 2.2.3 of [35] rely on our Theorem 5.2 which is only established for n even.Hence for real C*-algebras, we need to take m to be even in Lemma 2.2.1 and n to be even in Lemma 2.2.3.This is however, sufficient for all subsequent arguments.

Asymptotic Morphisms
We appropriate the following definition of an asymptotic morphism from Section 25.1 of [4].The other definitions in this section and the next are adapted from [35].(2) for all a, b ∈ A and all λ ∈ R, the following functions vanish in norm as t → ∞: We say that two asymptotic morphisms φ t and ψ t from A to B are equivalent if φ t (a) − ψ t (a) vanishes as t → ∞ for all a ∈ A. We say that φ t and ψ t are homotopic if there is an asymptotic morphism Φ t from A to C([0, 1], B) such that Φ t (a)(0) = φ t (a) and Φ t (a)(1) = ψ t (a) for all a ∈ A. Equivalent asymptotic morphisms are homotopy equivalent (see Remark 25.1.2 of [4]).
We leave the easy proof of the next lemma to the reader.
Lemma 8.2.If A and B are real C*-algebras and φ is an asymptotic morphism from A to B, then there is an asymptotic morphism φ C : It can be proven, then, from the same result in the complex case, that for any asymptotic morphism φ we have lim sup t→∞ φ t (a) ≤ a for all a ∈ A (see Proposition 25.1.3 of [4]).Thus, an asymptotic morphism {φ t } gives rise to a unique homomorphism defined in the natural way; and every such homomorphism represents an asymptotic morphism, unique up to equivalence.
Lemma 8.3.Let A be separable and nuclear.Every asymptotic morphism from A to B is equivalent to one that is completely positive and contractive.Furthermore, if φ and ψ are homopic completely positive and contractive asymptotic morphisms from A to B, then in fact there is a homotopy from φ to ψ consisting of completely positive and contractive asymptotic morphisms.
Proof.Let φ be an asymptotic morphism from A to B. Then by Proposition 1.1.5 of [35], the complexification φ C is equivalent to an asymptotic morphism ψ that is completely positive and contractive.The map α : B C → B defined by α(a + ib) = a is completely positive and contractive.Then the restriction of α•ψ to A is a completely positive, contractive asymptotic morphism from A to B and is equivalent to φ.
The same construction can be applied to a homotopy to prove the second statement.
Definition 8.4.Let φ and ψ be asymptotic morphisms from A to K R ⊗ D.
We define an asymptotic morphism φ ⊕ ψ, also from A to K R ⊗ D, as follows.
Choose an isomorphism δ : Lemma 8.5.The asymptotic morphism φ ⊕ ψ is well defined up to unitary equivalence, as well as up to homotopy.
Proof.As in the complex case every automorphism of K R is implemented by a unitary in U(B(H R )) (the proof in, for example, Lemma V.6.1 of [20] works in the real case).Furthermore, by [36], U(B(H R )) is path connected.(In fact, by Theorem 3 of [29], it is contractible.)Definition 8.6.Let φ : A → B be an asymptotic morphism of real C*-algebras and let p ∈ A be a projection.A tail projection for φ(p) is a continuous path p t of projections for t ∈ [0, ∞) such that lim t→∞ φ t (p) − p t = 0. We say that φ is full if there is a full projection p ∈ A such that φ(p) has a full tail projection.
Definition 8.7.Let A and B be real C*-algebras.Two asymptotic morphisms φ and ψ from A and B are asymptotically unitarily equivalent if there is a continuous family of unitary elements u t ∈ B such that lim t→∞ u t φ t (a)u * t − ψ t (a) = 0 for all a ∈ A.
With these definitions, all the results of Sections 1.2 and 1.3 of [35] hold for real C*-algebras.Theorem 8.9 (Theorem 2.3.7 of [35]).Let A be a separable, nuclear, unital, and c-simple.Let D 0 be a unital C*-algebra, and let D = O R ∞ ⊗ D 0 .Then two full asymptotic morphisms from A to K R ⊗ D are asymptotically unitarily equivalent if and only they are homotopic.
Proof.The proof of Theorem 2.3.7 in [35] as well as the proofs of all of the preceeding lemmas in Section 2.3 of [35] can be proven in the real case with the same proofs, with some extra attention paid to the issue of connectedness of unitary groups.In a few places Phillips uses the fact that the unitary group of O 2 is connected.
2 ) ∼ = 0.However, on page 85 of [35], Phillips also uses the fact that the unitary group of a corner algebra of O ∞ is connected.The corresponding statement in the real case is not true since We will show how to adjust the proof so that it works in the real case.At this point in the proof we are (using Phillips' notation) trying to find a path of partial isometries from w n + f n+2 to v n+1 + w n+1 (these are partial isometries from f n+1 + f n+2 to f n+2 + e).If the unitaries (w n + f n+2 ) * (v n+1 + w n+1 ) and f n+1 + f n+2 are not in the same connected component of (f n+1 + , then this can be changed by by multiplying w n+1 on the right by a suitable unitary in f n+2 O R ∞ f n+2 .Thus by re-choosing the w n 's inductively, we can be sure that there is an appropriate path of partial isometries at each step.9. Groups of Asymptotic Morphisms Definition 9.1.Let A be a real, separable, nuclear, unital, c-simple C*-algebra and let D be unital.We define E A (D) to be the the set of homotopy classes of full asymptotic morphisms from More generally, for D unital or not, we define Proposition 9.2.Let A be real, separable, nuclear, unital, and c-simple.Then E A (−) is a functor from the category of separable real C*-algebras with homotopy classes of asymptotic morphisms to abelian groups, that is homotopy invariant, stable, half exact, and split exact.
Proof.In the complex case, these results are proven in Section 3.1 of [35].In the real case, they are proven the same way.Note that split exactness follows from homotopy invariance and half exactness by Corollary 3.5 of [12].Lemma 9.3.Let A and B be C*-algebras (real or complex).Let φ : A → B be an asymptotic morphism.If p, q are projections in A with p ≤ q, then there are tail projections p t (for φ(p)) and q t (for φ(q)) in B with p t ≤ q t for all t.
Proof.Let p t and q t be arbitrary tail projections corresponding to φ(p) and φ(q), respectively (these exist as in Remark 1.2.2 of [35]).One can easily show that lim t→∞ p t − q t p t q t = 0 .
For each t, the element q t p t q t is a self adjoint and asymptotically idempotent element of q t Bq t .Therefore, there is a continuous path of projections p t ∈ q t Bq t such that lim t→∞ q t p t q t − p t = 0 .
The tail projections p t and q t have the desired properties.
We note that if A and D are complex C*-algebras there are two groups one might consider: we let E C A (D) denote the functor of [35] that is based on complex asymptotic morphisms.On the other hand, according to the notation established in Definition 9.1, the asymptotic morphisms comprising E A (D) are only required to be asymptotically linear over R (thus the complex structures of A and D are forgotten).The following theorem relates the two groups.
Proposition 9.4.If A is a real C*-algebra satisfying the hypotheses of Definition 9.1 and D is a complex unital C*-algebra, then there is a isomorphism which is natural with respect to complex homomorphisms.
Proof.We show that for a real unital C*-algebra A and a complex C*-algebra B, there is a bijection + of equivalence classes of full asymptotic morphisms.Given a complex asymptotic morphism φ from A C to B, then we let Γ(φ) be the restriction of φ to A. If φ is full, then we claim that Γ(φ) is full.Since φ is full, there is a full projection p ∈ A C and a full tail projection r t ∈ B such that φ t (p) − r t → 0. Applying Lemma 9.3 to p ≤ 1 we obtain tail projections p t and q t for p and 1, respectively, such that p t ≤ q t for all t.Since the tail projections p t and r t are asymptotically equal, it must be that p t are full projections.It follows that q t are also full projections; and since they are tail projections for the full projection 1 A in A, it follows that Γ(φ) is full.Given a real asymptotic morphism ψ from A to B, then defines a complex asymptotic morphism from A C to B. Suppose that ψ is full.Let p be a full projection in A and let q t ∈ B be a full tail projection for ψ(p).Then clearly p is full in A C and q t is a full tail projection for ∆(ψ(p)).Hence ∆(ψ) is full.It is immediate that ∆ is a two-sided inverse for Γ.Furthermore, in the case that B is stable, it is easy to see that Γ preserves the semigroup operation of Definition 8.4.Therefore, under the hypotheses of the theorem, there is an group isomorphism E A (D) ∼ = E C A C (D).
Proposition 9.5.Let A be a separable, nuclear, c-simple unital, real C*algebra.Let B be a separable real C*-algebra.Then there is a natural isomophism KK(A, B) ∼ = E A (B).
The proof in the complex case takes place in Section 3.2 of [35].Rather than reconstructing all of the arguments in the real case, we give a proof that uses results from [12] to reduce the real case to the complex case.
Proof of Proposition 9.5.Fix A satisfying the hypotheses above.Let e be a rank one projection in K R and let ι A : is an isomorphism for all separable real C*-algebras B. By Theorem 3.9 of [12] it suffices to show that α is an isomorphism when B is complex.In the complex case we have the element [ [24] there is a unique natural transformation A special case of Theorem 3.2.6 of [35] shows that α C is an isomorphism for all separable complex C*-algebras B. Consider the following diagram for a complex C*-algebra B, where µ is the isomorphism of Proposition 9.4 above and ν is the isomorphism of Lemma 4.3 of [9].To complete the proof, we only need to show that the diagram commutes.Since the homomorphism α C is characterized by the value of α C ([1 A C ]) it suffices to consider the case B = A C as in the diagram From the construction of ν in the proof of Lemma 4.3 of [9] it is apparent that ν([ On the other hand, it is apparent from the construction of µ in the proof of Proposition 9.4 above that µ The following is the real version of Theorems 4.1.1 and 4.1.3 of [35].
Theorem 9.6.Let A be a real separable unital nuclear c-simple C*-algebra and let D be a separable unital C*-algebra.Then the following groups are naturally isomorphic, via the obvious maps.
(1) KK(A, D) (2) The set of asymptotic unitary equivalence classes of full homomorphisms from ) The set of asymptotic unitary equivalence classes of full homomor- Proof.The proof of the isomorphism of ( 1), (2), and (3) is the same as the proof of Theorem 4.1.1 in [35].The proof of the isomorphism of ( 1), (4), and (5) relies on Lemma 9.7 below (which is the real version of Lemma 4.1.2 of [35]).Once that lemma is established, the proof of the isomorphism of ( 1), (4), and ( 5) is the same as the proof of Theorem 4.1.3 of [35].
Lemma 9.7.Let A be separable, nuclear, unital, and c-simple; let D 0 be separable and unital; and let Then there is an asymptotic unitary equivalence from φ to ψ that consists of unitaries in U 0 ((K R ⊗ D) + ).The proof will be essentially the same as the proof of Lemma 4.1.2 of [35].However, that proof has an error in the third paragraph.The element w t introduced there does not seem to be a unitary as purported.Also, the order of the product in the definition of z t seems wrong.Fortunately, there is an easy fix and most of the proof can be left as it is.For clarity and completeness we present the entire proof, but the only significant difference is the unitary w in the third paragraph and following.In places where the proof does not change (such as the entire first and second paragraphs, and most of the final paragraph), we use exactly the same language as in [35], except for the references to previous results in the present paper.
Proof of Lemma 9.7.Let {e ij } be a system of matrix units for K R .Identify A with the subalgebra e 11 ⊗ A of K R ⊗ A. Define ψ (0) t and ψ (0) to be the restrictions of φ t and ψ to A. Then [φ (0) 0 ] = [ψ (0) ] in KK 0 (A, D).It follows from (the equivalence of ( 1) and (3) of) Theorem 9.6 that φ (0) 0 is homotopic to ψ (0) .Therefore φ (0) 0 and ψ (0) are homotopic as asymptotic morphisms, and Theorem 8.9 provides an asymptotic unitary equivalence and such that c is homotopic to u −1 0 .Then c commutes with every ψ (0) (a).Replacing u t by cu t , we obtain an asymptotic unitary equivalence, which we again call t → u t , from φ (0) to ψ (0) which is in U 0 (K R ⊗ D) + ).Define e ij = e ij ⊗ 1.Then in particular u t φ t (e 11 )u * t → ψ(e 11 ) as t → ∞.Therefore there is a continuous path t → z (1) t ∈ U 0 ((K R ⊗D) + ) such that z In a neighborhood of each t, all but finitely many of the z (k) t are equal to 1, so this limit of products yields a continuous path of unitaries of U 0 ((K R ⊗ D) + ).Moreover, z t φ t (e ij )z * t = ψ(e ij ) whenever t ≥ i, j, so that lim t→∞ z t φ t (e ij )z * t = ψ(e ij ) for all i and j, while t ) * = ψ(e 11 ⊗ a) for all a ∈ A. Since the e ij and e 11 ⊗ a generate K R ⊗ A, this shows that t → z t is an asymptotic unitary equivalence.

Classification of Real Kirchberg Algebras
We now present our main classification theorems for real Kirchberg algebras, analogous to the results of Section 4.2 of [35].
Theorem 10.1.Let A and B be unital separable nuclear purely infinite c-simple C*-algebras.
(1) Let η be an invertible element in KK(A, B).Then there is an isomorphism Then there is an isomorphism φ : A → B such that [φ] = η.
Proof.As in the proofs of Theorem 4. (1) Let A be a complex C*-algebra.A real form of A is a real C*-algebra B such that Corollary 10.5.Let A be a complex unital separable nuclear purely infinite simple C*-algebra satisfying the universal coefficient  [35], implies that B is a real form of A.

Real Forms of Cuntz Algebras
In this section, we use Corollary 10.5 to give a complete description of all real forms of the complex Cuntz algebras O n for n ∈ {2, . . ., ∞}.The natural real form of O n is the real Cuntz algebra O R n , but we will find that there are others when n is odd.For reference, we show in Table 1 the groups making up K CRT (O R n ).In the case of n = ∞ this arises from the isomorphism K CRT (R) ∼ = K CRT (O R ∞ ) of Proposition 2.2; while for finite n, these CRT-modules were computed in Section 5.1 of [8].
(1) For n even or n = ∞, there is up to isomorphism only one real form of O n : the real Cuntz algebra O R n .
(2) For n odd, there are up to isomorphism two real forms of O n : the real Cuntz algebra O R n and an exotic real form E n .
Proof.First check that for odd integers n, n ≥ 3, the groups and operations shown in Table 2 form an acyclic CRT-module.Using Corollary 10.3 (that is, Theorem 1 of [10]), let E n be the unique real unital separable nuclear csimple purely infinite C*-algebra satisfying the universal coefficient theorem with united K-theory as shown in Table 2 and such that [1 En ] corresponds to a generator of the group in the real part in degree 0. By Corollary 10.5, the problem of classifying real forms of O n (for n ∈ {2, 3, . . ., ∞}) reduces to the algebraic problem of classifying real forms of (K * (O n ), [1 On ]).Suppose that (M, m) is such a real form.For n even (respectively n = ∞) we will show that (M, m) is isomorphic to For n odd we will show that (M, m) is either isomorphic to (K CRT

Proposition 2 . 3 (
Corollary 4.11 of [9]).Let A and B be real separable C*algebras such that A C and B C are in N .Then A and B are KK-equivalent if and only a is an element of A, then by definition the spectrum σ(a) is found by passing to A C .)We begin by making an explicit mention of a fairly well-known result about real simple C*-algebras.Definition 3.1.A real C*-algebra A is c-simple if A C is simple.Proposition 3.2.A simple real C*-algebra A is either c-simple or is isomorphic to a simple complex C*-algebra.Proof.Let I be a proper ideal in A C .Then J = A ∩ I ∩ Φ(I) = 0 and so I ∩ Φ(I) = 0. Furthermore, I + Φ(I) = A C .It then follows that the map x → x + Φ(x) is an isomorphism from the complex C*-algebra I onto A.

Corollary 3. 10 . ( 1 ) 2 )
If A and B are stably isomorphic real C*-algebras, and if A is purely infinite and c-simple then so is B. (Any inductive limit of real purely infinite c-simple C*-algebras is again purely infinite and c-simple.(3) If A and B are purely infinite and c-simple, then so is A ⊗ min B.

Lemma 4 . 5 .
Let A be a real unital C*-algebra.If every unitary u ∈ U 0 (A) can be connected to the identity by a rectifiable path of length no more than M , then cel(A) ≤ M .Definition 4.6.A real C*-algebra A has real skew rank zero if the elements of A sk with finite spectrum are dense in A sk .

Proposition 4 . 7 .
Let A be a real unital c-simple purely infinite C*-algebra satisfying [1] ∈ 2K 0 (A).Then the invertibles of A sk are dense in A sk and A has real skew rank zero.
there is a partial isometry s such that s * s = 1 − p and ss * = p.Since f (a)g(a) = 0 we have f (a) = (1 − p)f (a)(1 − p).Let b = f (a) + ε(s − s * ).In matrix form under the decomposition indicated by the projection sum 1 = (1 − p) + p we have b = f (a) −ε ε 0 whence b is invertible.This proves the first statement.
For each k ∈ {1, . . ., n} let c k1 ∈ A C be a partial isometry such that c * k1 c k1 = p k1 and c k1 c * k1 < p k1 .Thenc k2 = Φ(c k1 ) satisfies c * k2 c k2 = p k2 and c k2 c * k2 < p k2 and c k = c k1 +c k2 ∈ A satisfies c * k c k = p k1 +p k2 and c k c * k < p k1 +p k2 .For k = 0 we obtain partial isometries c 0k ∈ A such that c * 0i c 0i = p 0i and c ki c * ki < p ki .Then c = p + c ki ∈ A satisfies c * c = e 1 , cc * = e 1 − (p ki − c ki c * ki ), and cuc * = u 0 + λ ki c ki c * ki .Similarly we can find a collection of partial isometries d ki with domain projection q ki and range projection a subprojection of p ki − c * ki c ki that also satisfy Φ(d k1 ) = d k2 for k = 0 and Φ(d ki ) = d ki for k = 0. Then the partial isometry d = d ki ∈ A satisfies d * d = e 2 + e 3 , dd * ≤ (p ki − c ki c * ki ), and d λ ki q ki d * = λ ki d ki d * ki .Now, choose a partial isometry b such that b * b < e 4 , bb * = (p ki − c ki c * ki − d ki d * ki ) and define w 0 = λ ki b * (p ki − c ki c * ki − d ki d * ki )b .Then z 0 = b + c + d is a partial isometry with z * 0 z 0 = e 1 + e 2 + e 3 + b * b and z 0 z * 0 = e 1 .So in K 0 (A) we have [e 1 ] = [e 1 + e 2 + e 3 + b * b], which implies [1 − e 1 ] = [e 4 − b * b].By Proposition 11 of [10], there is a partial isometry z 1 ∈ A such that z 1 z * 1 = 1 − e 1 and z * 1 z 1 = e 4 − b * b.Then w = w 0 + e 4 − b * b is a unitary with finite spectrum in e 4 Ae 4 and z

(nj −1) j = 0
Documenta Mathematica 16 (2011) 619-655 and let A ⊗ γ B be the real C*-algebra obtained by completion.Then the complexification (A ⊗ γ B) C has a unique C*-norm extending that on A ⊗ R B. Thus every C*-norm on the algebraic tensor product A ⊗ R B extends uniquely to a C*-norm on A C ⊗ C B C .Part (2) follows immediately from this claim.

Theorem 7 . 1 .
Let D be a real unital purely infinite simple C*-algebra, and let φ, ψ : O R ∞ → D be unital homomorphisms.Then φ is asymptotically unitarily equivalent to ψ.

Definition 8 . 1 .
Let A and B be real C*-algebras.An asymptotic morphism φ from A to B is a family {φ t } t∈[0,∞) of maps φ t : A → B such that (1) the map t → φ t (a) is continuous for each a ∈ A, and Documenta Mathematica 16 (2011) 619-655

Definition 8 . 8 .
Let A and D be real C*-algebras.An asymptotic morphism φ : A → D has a standard factorization through O R ∞ ⊗A if there is an asymptotic morphism ψ : O R ∞ ⊗ A → D such that the asymptotic morphisms φ(a) and ψ(1 ⊗ a) (both from A to D) are asymptotically unitarily equivalent.Similarly, φ is asymptotically trivially factorizable if there is an asymptotic morphism ψ : O R 2 ⊗ A → D such that φ(a) and ψ(1 ⊗ a) are asymptotically unitarily equivalent.

2 . 1 and 1 ) 2 ) 3 ) 5 . 10 . 3 . ( 1 ) 2 )
Corollary 4.2.2 of [35].Theorem 10.2.Let A and B be unital separable nuclear purely infinite c-simple C*-algebras that satisfy the universal coefficient theorem.(The stable C*-algebras K R ⊗ A and K R ⊗ B are isomorphic if and only if K CRT (A) and K CRT (B) are isomorphic CRT-modules.(The unital C*-algebras A and B are isomorphic if and only if the invariants (K CRT (A), [1 A ]) and (K CRT (B), [1 B ]) are isomorphic.(The stable C*-algebras K R ⊗ A and K R ⊗ B are isomorphic if and only if K CR (A) and K CR (B) are isomorphic CR-modules.(4) The unital C*-algebras A and B are isomorphic if and only if the invariants (K CR (A), [1 A ]) and (K CR (B), [1 B ]) are isomorphic.Proof.Parts (1) and (2) are proven as in the proof of Theorem 4.2.4 of [35], using Proposition 2.3.Parts (3) and (4) then follow by Proposition 2.Corollary The functor A → K CRT (A) is a bijection from isomorphism classes of real stable separable nuclear purely infinite c-simple C*-algebras that satisfy the universal coefficient theorem to isomorphism classes of countable acyclic CRT-modules.(The functor A → (K CRT (A), [1 A ]) is a bijection from isomorphism classes of real unital separable nuclear purely infinite c-simple C*algebras that satisfy the universal coefficient theorem to isomorphism classes of countable acyclic CRT-modules M with distinguished element m ∈ M O 0 .Proof.Combine Theorem 10.2 above with Theorem 1 of [10].Definition 10.4.

( 1 )
The functor B → K CRT (B) is a bijection from isomorphism classes of real forms of K R ⊗ A to isomorphism classes of real forms of K * (A).(2)The functor B → (K CRT (B), [1 B ]) is a bijection from isomorphism classes of real forms of A to isomorphism classes of real forms of(K * (A), [1 A ]). Proof.If B is a real form of K R ⊗ A, then B is necessarily stable separable nuclear purely infinite and c-simple.Then KU * (B) = K * (B C ) ∼ = K * (A), so K CRT (B) is a real form of K * (A).Conversely, suppose M is a real form of K * (A).Since K * (A)is countable, the exact sequences of Section 2.3 of[14] imply that M is countable.Then by Corollary 10.3, M ∼ = K CRT (B) for some real stable separable nuclear purely infinite c-simple C*-algebra satisfying the universal coefficient theorem.Since K * (B C ) ∼ = K * (A), it follows from Theorem 4.2.4 of[35] that B C ∼ = A hence B is a real form of A. Furthermore, Corollary 10.3 also implies that B is unique up to isomorphism.In the unital case, suppose that B is a real form of A. As there is a isomorphismB C ∼ = A and the unit of B C is c(1 B there is an isomorphism φ : KU * (B) → K * (A) such that φ * (c([1 B ])) = [1 A ]. Thus (K CRT (B), [1 B ]) is a real form of (K * (A), [1 A ]). Conversely, if (M, m) is a real form of K * (A),then let B be a real unital separable nuclear purely infinite c-simple C*-algebra such that (K CRT (B), [1 B ]) ∼ = (M, m).Again, Theorem 4.2.4 of

1 Proposition 2 . 5 2 ∼ 2 ∼ 2 ∼
it suffices to restrict our attention to the CR-module consisting of the real and complex parts of M .Since (M, m) is a real form of (K* (O n ), [1 On ]) we know that M U 0 ∼ = Z n−1 (respectively M U 0 ∼ = Z when n = ∞), M U 1 = 0, and m ∈ M O 0 .We further suppose that c 0 (m) ∈ M U 0 is a generator (corresponding to the class of the unit in K 0 (O n )).We will compute the real part of M (and the behavior of the operations η O , ξ, r, c, ψ U ) using the long exact sequence• • • → M → M O n−1 → .. .and the CRT-relations described in Section 2. Since M U k = 0 for k odd it follows that (η O ) k is injective for k odd and surjective for k even.Furthermore, our hypothesis that c 0 (m) generates M U 0 implies that c 0 is surjective, which implies that r −2 = 0 and that (ηO ) −2 is injective.Thus (η O ) −2 : M O −2 → M O −1 is an isomorphism and η 3 O : M O −3 → M O 0 is injective.Then the relations η 3 O = 0 and 2η O = 0 imply that M O −3 = 0 and that M O −2 consists only of 2-torsion.Suppose first that M O −= M O −1 = 0.Then using the long exact sequence above and the relation rc = 2, the rest of the groups of M O can be easily computed; except that in the case that n is odd we encounter an extension problem wherein M O 2 is either isomorphic to Z 4 or to Z 2 ⊕ Z 2 .In that case, the same argument as in the computation of K CRT (O R n ) in Section 5.1 of[8] shows that M O = Z 4 exactly when n − 1 ≡ 0 (mod 4) and M O = Z 2 ⊕ Z 2 exactly when n − 1 ≡ 2 (mod 4).Thus we find that the real and complex parts of M (as well as the operations η O ξ, r, c, ψ U ) are isomorphic to the real and complex parts ofK CRT (O R n ) (respectively K CRT (O R ∞ )).Documenta Mathematica 16 (2011) 619-655 [31]), let a, b be non-zero elements of A, identified with e 11 (K R ⊗ A)e 11 .We are assuming A C is simple, so Theorem 2.4 of[17]applied to the unital algebra pAp implies that K ⊗ pA C p is algebraically simple.Then by Proposition 3.5, K⊗A C is algebraically simple, whence K R ⊗A is.The argument from (ii) ⇒ (xi) of Theorem 2.2 of[31], then produces x, y ∈ K R ⊗ A with a = xby, so a = (e 11 xe 11 )b(e 11 ye 11 ).For (2) ⇒ (1), we use a simplified argument based on the proof of Theorem 1.2 of[31].Note first that if a nonzero projection can be found in A then Lemma 3.7 gives the result.(In particular, this takes care of the unital case.)Let a, d be non-zero positive elements of A with da = ad = a (for a positive element x with norm 1 take a = f 1/2 (x) and d = f 1/4 (x)).Then let s, t ∈ A with d = sat and let y = (as * sa) 1/2 t.An easy argument shows that |y||y * | = |y * R 2 by Lemma 6.2 and Proposition 6.3.Then by Theorem 5.2 we have κ • γ ≈ u 1 O R 2 and by Lemma 6.4 we have γ • κ ≈ u 1 A⊗O R 2 .Therefore, by (the real analog of) Corollary 2.3.4 of [39], separable, unital, and nuclear.There is a unital homomorphim γ : O R 2 → A ⊗ O R 2 given by x → 1 ⊗ x and there is a unital homomorphism κ : A ⊗ O R 2 → O (O R n ), [1 O R n ]) or to (K CRT (E n ), [1 En ]).Furthermore, by Documenta Mathematica 16 (2011) 619-655

Table 2 .
K CRT (E n ), for n odd and n ≥ 3.